Questions: Phase and Amplitude in Forced Oscillations

Question 1

An engineer wants to maximize the power delivered by a periodic driving force to a mechanical oscillator. At what condition is power transfer maximized?

• A. When the driving force and the oscillator's displacement are in phase (φ = 0°)
• B. When the driving force and the oscillator's velocity are in phase, which occurs when displacement lags force by 90°
• C. When the driving force and the oscillator's velocity are 180° out of phase
• D. When the driving frequency ω is much higher than the natural frequency ω₀

Answer: (B)

Power delivered by a force equals force times velocity (P = F·v). This is maximized when force and velocity are in phase — they point in the same direction simultaneously. At resonance (ω = ω₀), displacement lags the driving force by 90°. Since velocity is the time derivative of displacement, a 90° lag in displacement means velocity is exactly in phase with force. This is why resonance maximizes power input: not because of any special alignment of force and displacement, but because the 90° lag in displacement produces a 0° lag between force and velocity. Option A (force and displacement in phase) would mean velocity leads force by 90°, which actually minimizes power transfer.

Question 2

What happens to the phase lag between driving force and oscillator displacement as the driving frequency increases from far below to far above the natural frequency ω₀?

• A. Phase lag stays near 0° throughout, since the oscillator tracks the force at all frequencies
• B. Phase lag stays near 90° throughout, since resonance dominates the response
• C. Phase lag increases continuously from near 0° at low frequencies to near 180° at high frequencies, passing through exactly 90° at resonance
• D. Phase lag jumps abruptly from 0° to 180° at the resonant frequency

Answer: (C)

The phase behavior sweeps continuously: at low driving frequencies (ω << ω₀), the oscillator has time to follow the force nearly instantaneously, so φ ≈ 0°. At ω = ω₀, the lag is exactly 90°. At high driving frequencies (ω >> ω₀), the oscillator cannot keep up and lags by nearly 180° — moving almost opposite to the driving force. This continuous sweep from 0° to 180° is a universal feature of driven oscillators and is more diagnostically useful than the amplitude curve alone for identifying resonance.

Question 3

The 90° phase lag at resonance is directly responsible for large oscillation amplitudes, because it brings the oscillator's velocity into phase with the driving force and maximizes the rate of energy input.

• T. True
• F. False

Answer: True

This is the mechanistic explanation for why resonance produces large amplitude. Power = F·v is maximized when force and velocity are in phase. At resonance, the 90° displacement lag means velocity is in phase with force (since v = dx/dt, and differentiating a cosine delayed by 90° gives a sine, which aligns with the cosine driving force). Sustained maximum power input drives the amplitude to its peak, limited only by damping. Without this phase relationship, the energy input would be partially cancelled each cycle.

Question 4

At very low driving frequencies (ω << ω₀), the oscillator significantly lags behind the driving force because its inertia prevents it from responding to slow oscillations.

• T. True
• F. False

Answer: False

At low driving frequencies, the opposite is true: the oscillator has plenty of time to follow each slow oscillation of the driving force and responds nearly in phase (φ ≈ 0°). Inertia matters at high frequencies, where the oscillator cannot accelerate fast enough to keep up with rapid direction changes — that is when the phase lag approaches 180°. The intuition is like slowly pushing a child on a swing: if you push gently and slowly, they follow your lead easily. It is the fast-oscillation case where lag becomes large.

Question 5

Explain, using the relationship between force, velocity, and power, why the 90° phase lag at resonance causes the oscillator's amplitude to grow large.

Model answer: The time-averaged power delivered to an oscillator is ⟨P⟩ = (F₀Aω/2)sin(φ), where φ is the phase lag between the driving force and displacement. This is maximized when sin(φ) = 1, i.e., φ = 90°. At resonance, displacement lags the driving force by exactly 90°, which means velocity — the time derivative of displacement — is exactly in phase with the force. Force and velocity pointing in the same direction at every instant means the driving force always does positive work on the oscillator. This sustained maximum power input drives the amplitude to its resonant peak, limited only by the rate at which damping dissipates energy.

The 90° phase lag is not just a coincidental feature of resonance — it is the mechanism that makes resonance energetically special. At any other phase lag, sin(φ) < 1, meaning some fraction of the driving force's work is cancelled each cycle. Only at φ = 90° is every joule of work done by the driving force pumped into the oscillator. For lightly damped systems, this produces very large amplitudes before a steady state is reached between energy input and dissipation.